Okay, so this is a non-standard problem. The game is to play tennis with theorems. Gameplay goes as follows :
(1) A theorem comes your way.
(2) You respond to the theorem with another theorem, which you must relate to the first.
If you play, try to keep the game open, by stating theorems which can easily be related to by others.
I'll begin with a well-known theorem from high-school geometry : the medians of a triangle intersect.
I'm guessing perhaps he meant you can use parts of the previously posted theorem in your one.
Like if I said
The cardinality of the integers is
Your next theorem could be about cardinality, the integers or something to do with .
Is that what's going on in here..?
(Although that does seem a little simple for a maths challenge question...)
Absolutely right! Make all necessary assumptions.
Tonio : here's how I thought it related to the geometric theorem. The point of intersection of the medians is the center of gravity of the three vertices, assuming they have equal mass. With this in mind, the theorem you stated is a specific case of the following more general theorem : the center of mass of points of equal (nonzero ) mass, of which are blue and red, lies of the way along the segment joining the centers of mass of the red points and of the blue points. But we can also view the points as events in a real probability space, with the expectation replacing the center of mass...
Deadstar : that's essentially it. The point is just to have fun and to see what kind of unexpected theorems people will pull out of their hats.
Alright!
This one is from Putnam a few years back. Let be a finite collection of open discs in the plane (possibly overlapping). Then there is a subcollection of them, such that they are pairwise disjoint, and such that covers , where denotes the disc with the same center as but with three times the radius.